Singular measures - Encyclopedia of Mathematics
WEBAug 18, 2012 · For general, i.e. non $\sigma$-finite (nonnegative) measures, the concept can be generalized in the following way: $\mu$ and $\nu$ are singular if the only (nonnegative) measure $\alpha$ on $\mathcal{B}$ with the property \[ \alpha (A)\leq \min \{\mu (A), \nu (A)\} \qquad \forall A\in\mathcal{B} \] …
Chapter 7 Product Measures | Measure theory lecture notes
WEBIn this section we look at taking two measure spaces \((E, \mathcal{E}, \mu)\) and \((F, \mathcal{F}, \nu)\) and defining a \(\sigma\) algebra and a measure on the product space \(E \times F\). This will give us another way of defining Lebesgue measure on \(\mathbb{R}^d\) .
Uniqueness of product measure (non $\\sigma$-finite case)
WEBCan the product of a $\sigma$-finite measure with a non-$\sigma$-finite one be a $\sigma$-finite measure?
Important examples of measures which are not $\\sigma$-finite
WEBConsider ${\Bbb R}^2$ with a $\sigma$-algebra $\Sigma$ generated by smooth curves of finite length. Construct a measure $\mu$ so that $\mu(\gamma) = $length of $\gamma$ if $\gamma$ is a curve. This is a measure which is not $\sigma$-finite.
What is a non linear $\sigma$ model? - Physics Stack Exchange
WEBWhat exactly is a non linear $\sigma$ model? In many books one can view many different types of non linear $\sigma$ models but I don't understand what is the link between all of them and why it is ...
Example of an algebra which is not a σ-algebra.
WEBNov 9, 2012 · Let $X$ be an infinite set, and $\mathcal A$ be the collection of all subsets of $X$ which are finite or have finite complement. Then $\mathcal A$ is an algebra of sets which is not a $\sigma$-algebra.
measure theory - Fubini's theorem without completeness or $\sigma …
WEBWith Fubini on non-$\sigma$-finite spaces you have to be careful about what you mean by 'product measure'. Suppose we use the Carathéodory method to extend the premeasure from measurable rectangles to a complete measure.
Radon-Nikodym theorem for non-sigma finite measures
WEBWhen $\mu$ is sigma-finite, the Radon-Nikodym theorem provides a decomposition of $\lambda$ in a sum of an absolutely continuous measure wrt $\mu$ plus a singular measure wrt $\mu$. Question. Is that statement still true without the sigma-finiteness of …
Non-linear sigma model - Wikipedia
WEBIn quantum field theory, a nonlinear σ model describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. The non-linear σ -model was introduced by Gell-Mann & Lévy (1960, section 6), who named it after a field corresponding to a spinless meson called σ in their model. [1]
WEBnon-linear sigma model is a map from the Minkowski spacetime to a non-linear target space, namely a curved Riemannian manifold. In terms of the target space’s coordinates ˚a (a= 1;:::;N) and its metric ds2 = gab(˚)d˚ad˚b; (1) the non-linear sigma model comprises N scalar elds ˚a(x) with the purely-kinetic La-grangian L= 1 2 g ab ...
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